On finite strongly critical rings

Abstract

In the present paper, some properties of strongly critical rings are investigated. It is proved that every simple finite ring and each critical ring of order p² (p is a prime) are strongly critical. There is an example of critical ring of order 8 which is not strongly critical. It is also proved that if R is a finite ring and M_n (R) is a strongly critical ring, then R is a strongly critical ring. For rings with unity, it is proved that: 1) if R is a finite ring, R / J (R) = M_n (GF(q)) and J (R) is a strongly critical ring, then R is a strongly critical ring; 2) R is strongly critical ring iff M_n (R) is a strongly critical ring (for any n≥1).